Probability Q using Stirling's Formula

Suppose that there is a large group of people, consisting of exactly 2N women and 2N men. The group is split in half at random. What is the probability that each half contains exactly N women and N men?

And the second part asks for a given N, set up Stirling's Formula so you can calculate the approx value for a given N.

I am having trouble thinking how to set up this problem. I know as N gets bigger, the possible combinations increase....I'm thinking 2^n.
I also know that the order does not matter, but with each selection you are taking removing someone from the pool of 4N people.

Suppose that there is a large group of people, consisting of exactly 2N women and 2N men. The group is split in half at random. What is the probability that each half contains exactly N women and N men?

And the second part asks for a given N, set up Stirling's Formula so you can calculate the approx value for a given N.

I am having trouble thinking how to set up this problem. I know as N gets bigger, the possible combinations increase....I'm thinking 2^n.
I also know that the order does not matter, but with each selection you are taking removing someone from the pool of 4N people.

You want a _hypergeometric_ distribution here. The hypergeometric describes the following: given a population of M items, M1 of type 1 and M2 of type 2, suppose you extract a a sample of n items at random without replacement. What is the probability your sample contains exactly k items of type 1? In your case, M = 2N, M1 = M2 = N, n = N and k = N. There are standard formulas for the hypergeometric distribution.